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The polar form of a complex number and the roots of unity givesince $512=2^9$. Using the complex conjugate to compute the squared complex modulus,The residue $3j-k$ runs through every class modulo $9$, already with $j=0$. The cosine is even, so the nine angles give five distinct values, represented by $r=0,1,2,3,4$. They are distinct because cosine is strictly decreasing on $[0,\pi ]$ and these representative angles lie in that interval. HenceEvery listed value is attained by choosing $z_1=1$ and $z_2=2e^{2\pi ir/9}$.

Squaring first avoids unnecessary expansion: $(2+2i{)}^2=8i$, and squaring again gives the fourth power $\boxed{z^4=-64}$. In the polar form of a complex number, $z=2\sqrt{2}{e}^{i\pi /4}$; multiplying its complex argument by four gives the same negative-real result.

The number being squared has polar form of a complex number $e^{2\pi i/3}$. Squaring doubles its argument, giving $e^{4\pi i/3}=-1/2-i\sqrt{3}/2$. Hence

In polar form of a complex number, $1+i=\sqrt{2}{e}^{i\pi /4}$. By De Moivre's theorem,

Differentiate the polar form of a complex number using the product rule and chain rule:A second differentiation givesThe two contributions to $2\dot{r}\dot{\theta }$ arise separately from differentiating $r\dot{\theta }$ and from differentiating the moving exponential factor.

For an eigenvector $\mathbf{v}=(v_1,v_2,v_3{)}^{\mathsf{T}}$, the equation $A\mathbf{v}=\lambda \mathbf{v}$ readsA nonzero solution requires ${\lambda }^3=1$. Write the three roots of unity in the requested polar form of a complex number:Indeed ${\mathbf{v}}_k=(1,{\lambda }_k^2,{\lambda }_k{)}^{\mathsf{T}}$ satisfies all three equations. These eigenvectors are unnormalized; any nonzero scalar multiples are equally valid.

In polar coordinates, $x=r\cos \theta$, $y=r\sin \theta$, with $r=\sqrt{x^2+y^2}$. Thus the polar form of a complex number isHere $r=|z|\ge 0$ is its complex modulus; for $z\ne 0$, the complex argument $\theta$ is determined modulo $2\pi$. At $z=0$ the angle is undefined, although every value of $\theta$ gives the same point when $r=0$.